Welcome! This project explores the use of Physics-Informed Neural Networks to solve a tuberculosis ODE model.
This project presents a computational solution to a tuberculosis (TB) compartmental model using:
- Classical numerical solvers
- Physics-Informed Neural Networks (PINNs)
The PINN approach integrates the system of ODEs into the neural network training process, allowing for continuous and accurate approximations of disease dynamics over time.
We modeled the spread and progression of TB using a system of four ODEs representing the following compartments:
- S(t): Susceptible individuals
- E(t): Exposed (infected but not yet infectious)
- I(t): Infectious individuals
- L(t): Latent or partially treated individuals
Two solution methods were implemented:
- Numerical integration methods (e.g., Runge-Kutta-Fehlberg, Heun’s Method, Gauss Quadrature)
- Physics-Informed Neural Network (PINN) trained using PyTorch
- Heun's Method (Predictor-Corrector)
- Classical Runge-Kutta 4th Order (RK4)
- Adaptive Runge-Kutta Method
- Physics-Informed Neural Networks (PINNs)
- Implemented using PyTorch
- Trained on ODE physics, initial conditions, and data points
- Supports extrapolation from t=0 to t=100
| Method | Most Accurate | Fastest Execution | Extrapolation Support |
|---|---|---|---|
| Heun | ✗ | Fast | ✗ |
| RK4 | ✓ Very Accurate | Moderate | ✗ |
| Adaptive RK | ✓ Accurate | Fastest | ✗ |
| PINN | ✓ Accurate | Fast (post-training) | ✓YES |
| Compartment | Heun | RK4 | Adaptive RK | PINN |
|---|---|---|---|---|
| S | 3.6878 | 0.0001 | 0.0163 | 1.37 |
| E | 3.3949 | 0.0001 | 0.0151 | 1.27 |
| I | 0.0227 | 0.0000 | 0.0001 | 0.009 |
| L | 0.000002 | 0.0000 | 0.0000 | 0.000001 |
- PINN outperformed numerical integration in speed (7900% faster).
📁 adaptive_quad/ → Contains adaptive_quad.py – Adaptive Quadrature integration method
📁 figures/ → Plots and result visualizations (MSE curves, method comparisons, etc.)
📁 gauss-methods/ → Contains gauss_methods.py – Gauss quadrature-based solver
📁 heun_non_self_start/ → Contains heun_non_self_start.py – Heun’s method for solving ODEs
📁 romberg_integration/ → Contains romberg_integration.py – Romberg integration-based solver
📁 runge-kutta-method/ → Contains runge_kutta_method.py – Classic 4th order Runge-Kutta method
Numerical_Project_Full.ipynb → Jupyter notebook version that includes all code and result analysis
Each Python script contains a solver implementation for one of the numerical methods used in this study, with consistent function structures and outputs and plots for modular testing and comparison
Live interactive summary hosted with GitHub Pages.
Romberg Integration is a numerical method used to estimate definite integrals with high precision. It combines the Trapezoidal Rule with Richardson Extrapolation to refine the accuracy of the result over successive approximations.
It starts from the basic trapezoidal estimate:
Tₙ = (h / 2) × [f(a) + 2 × ∑ f(xᵢ) + f(b)]
Then applies Richardson extrapolation recursively to eliminate lower-order error terms:
R(n, m) = [4ᵐ × R(n, m−1) − R(n−1, m−1)] / (4ᵐ − 1)
This builds a Romberg table where the bottom-right value is the most accurate approximation of the integral.
Although Romberg is powerful for integrating functions of a single variable (usually time), it is not suitable for our TB ODE model because:
- Our ODE system has the form:
dy/dt = f(y)
where f(y) does not depend on time t explicitly.
- Romberg Integration is most effective when integrating:
∫ f(t) dt
with function values changing significantly over t.
- In our model, since the right-hand side
f(y)is independent oft, the integration essentially becomes:
yₙ₊₁ ≈ yₙ + f(yₙ) × dt
which reduces to Euler’s method, but with extra computational cost and no added accuracy.
- [rowida moahemd ]
- [zyad hamed ]
- [Ali ahmed gad ]
- [yousef samy ]
- [yousef mortada ]
- [Mohamed ward ]
- [malak sherif ]
- [yomna sabry ]
- [mohamed nasser ]
- [ Abdelrahman Emad ]
- [farah yehia ]
- [yousef magdy]
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- Cairo University Numerical Methods materials (Spring 2025)